Hence, a Riemann sum approximation works backwards from a secant line approximation. Given an object's acceleration curve, a Riemann sum can be used to determine an object's velocity curve. Given an object's velocity curve for an object, a Riemann sum can be used to determine an object's position curve. In recognizable terms : In common words, acceleration is a measure of the change in speed of an object, either increasing acceleration or decreasing deceleration.
This definition is not completely accurate because it disregards the directional component of the velocity vector. Vectors have two components—magnitude and direction. When discussing speed, we only consider the change in magnitude. In conceptual terms: Acceleration is a quantity in physics that is defined to be the rate of change in the velocity of an object over time.
Since velocity is a vector, acceleration describes the rate of change in the magnitude and direction of the velocity of an object. When thinking in only one dimension, acceleration is the rate that something speeds up or slows down. In mathematical terms: Many different mathematical variations exist for acceleration.
Below is a partial listing:. In process terms : To compute the acceleration of an object, it is first essential to understand what type of motion is occurring. Once the type of motion is determined, a variety of mathematical equations can be applied, depending on the situation. Unfortunately, the acceleration is only easy to find in situations in which the object's motion is predictable. For instance, when an object is undergoing harmonic motion, the acceleration of the object can be determined because the object's position is predictable at any point in time.
In applicable terms : Any object in motion has acceleration. If the object's velocity is changing, the object is either accelerating or decelerating. If the object has constant velocity, the object's acceleration is zero. If an object is moving at a constant speed following a circular path, the object experiences a constant acceleration that points toward the center of the circle. Riemann sum: The approximation of the area of the region under a curve. Secant lines can be used to approximate the tangent to a curve by moving the points of intersection of the secant line closer to the point of tangency.
Pre-Lesson Assessment: Ask students the following questions to gauge their prior knowledge:. Formative Assessment: As students are engaged in the lesson, ask these or similar questions:. However, these contents do not necessarily represent the policies of the National Science Foundation, and you should not assume endorsement by the federal government. Why Teach Engineering in K? Find more at TeachEngineering.
Quick Look. Print this lesson Toggle Dropdown Print lesson and its associated curriculum. Suggest an edit. Discuss this lesson. Activities Associated with this Lesson Units serve as guides to a particular content or subject area. TE Newsletter. Subscribe to TE Newsletter. Summary Students observe four different classroom setups with objects in motion using toy cars, a ball on an incline, and a dynamics cart. At the first observation of each scenario, students sketch predicted position vs. Then the classroom scenarios are conducted again with a motion detector and accompanying tools to produce position vs.
If position is given by a function p x , then the velocity is the first derivative of that function, and the acceleration is the second derivative. By using differential equations with either velocity or acceleration, it is possible to find position and velocity functions from a known acceleration.
I want to talk about position, velocity and acceleration and how differential equations can be used to show the relationships between these. Imagine an object that's moving along a straight line. If we know that its position at any time t is s of t, here's it's position s of t, it's velocity v of t is the derivative of s of t and its acceleration is the derivative of velocity.
So if you think about what happens as we go in this direction, we're differentiating, right? Going downward we're differentiating. The derivative of position is velocity, the derivative of velocity is acceleration. So going in the reverse direction, and this is the way we're often going to go to have to go in.
Problems that we're going to do in this topic. We're going to have to anti-differentiate to go from acceleration to velocity, from velocity to position. In particular these equations can be used to model the motion of a falling object, since the acceleration due to gravity is constant. Calculus allows us to see the connection between these equations. First note that the derivative of the formula for position with respect to time, is the formula for velocity with respect to time.
Moreover, the derivative of formula for velocity with respect to time, is simply , the acceleration. Me Profile Supervise Logout. No, keep my work. Yes, delete my work. Keep the old version.
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